document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006338>Question 1006338</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve the problem<BR>\n" );
document.write( "two vertices of a triangle(5,-1) and (-2,3) and if the orthocentre lies on origin Find the third vertex:<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622493 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=622493\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the third vertex is at (-4,-7)\r<BR>\n" );
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document.write( "how to find it.\r<BR>\n" );
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document.write( "first understand what orthocenter means.<BR>\n" );
document.write( "it is the intersection of all the altitudes of the triangle.<BR>\n" );
document.write( "the altitudes of the triangle pass through one vertex and are perpendicular to the opposite side.\r<BR>\n" );
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document.write( "the rest if finding the equations of the lines of the triangles and of the altitudes of the triagnles and then finding the intersections of a couple of those lines to find the third point.\r<BR>\n" );
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document.write( "the specific application strategy is shown below:\r<BR>\n" );
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document.write( "first get the equation of the line through the points (-2,3) and (5,-1)\r<BR>\n" );
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document.write( "the equation of that line is y = (-4/7) * x + 13/7.\r<BR>\n" );
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document.write( "next find the line that is perpendicular to that line and passing through the point (0,0).\r<BR>\n" );
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document.write( "the equation of that line is y = (7/4) * x.\r<BR>\n" );
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document.write( "next find the equation of the line through the point (5,-1) and the point (0,0).\r<BR>\n" );
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document.write( "the equation of that line is y = (-1/5) * x.\r<BR>\n" );
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document.write( "next find the line that is perpendicular to y = (-1/5) * x and passing through the point (-2,3).\r<BR>\n" );
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document.write( "the equation of that line is y = 5 * x + 13.\r<BR>\n" );
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document.write( "next find the intersection of y = 5 * x + 13 with y = (7/4) * x.\r<BR>\n" );
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document.write( "that intersection is the point (-4,-7).\r<BR>\n" );
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document.write( "that's all you need.\r<BR>\n" );
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document.write( "as a bonus, i found the equation of the line through the points (-4,-7) and (5,-1) as well.\r<BR>\n" );
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document.write( "that equation is y = (2/3) * x - 13/3\r<BR>\n" );
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document.write( "the equation of the line through the origin and through the point (5,-1) is equal to y = (-3/2) * x.\r<BR>\n" );
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document.write( "all of this is shown on the following graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/112701.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the equations of the lines that form the sides of the triangle are shown in black.\r<BR>\n" );
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document.write( "the equations of the lines that are perpendicular to those lines and passing through the origin are shown in red.\r<BR>\n" );
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document.write( "basic concepts used:\r<BR>\n" );
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document.write( "slope intercept equation of a straight line is y = mx + b<BR>\n" );
document.write( "m is the slope<BR>\n" );
document.write( "b is the y-intercept.\r<BR>\n" );
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document.write( "slope is derived from 2 points on the line.<BR>\n" );
document.write( "first point is (x1,y1)<BR>\n" );
document.write( "second point is (x2,y2)<BR>\n" );
document.write( "slope = (y2-y1) / (x2-1)\r<BR>\n" );
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document.write( "slope of a line perpendicular to another line is the negative reciprocal of the slope of that line.<BR>\n" );
document.write( "if the slope of the line is -4/7, then the slope of the line perpendicular to it would be 7/4.\r<BR>\n" );
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document.write( "if you have the slope of a line and a point on that line, then replace x with the x-coordinate of the point and replace y with the y-coordinate of the point and then solve for b.<BR>\n" );
document.write( "example:<BR>\n" );
document.write( "y = mx + b.<BR>\n" );
document.write( "slope is 2/3.<BR>\n" );
document.write( "y = mx + b becomes y = (2/3) * x + b.<BR>\n" );
document.write( "point is (5,-1).<BR>\n" );
document.write( "y = (2/3) * x + b becomes -1 = (2/3) * 5 + b.<BR>\n" );
document.write( "simplify to get -1 = 10/3 + b.<BR>\n" );
document.write( "subtract 10/3 from both sides of the equation to get:<BR>\n" );
document.write( "-1 - 10/3 = b.<BR>\n" );
document.write( "solve for b to get b = -13/3.<BR>\n" );
document.write( "y = (2/3) * x + b becomes y = (2/3) * x - 13/3.\r<BR>\n" );
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document.write( "if you are confused by how any of this was derived, let me know and i'll do my best to clear up the confusion.\r<BR>\n" );
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